Let R be a ring with unity, GL (R) n the group of all invertible n by n matrices over R andGE (R) n the subgroup of GL (R) n generated by invertible elementary n by n matrices over R. If F is a field,then, in linear algebra, GL (F) GE (F) n n for every positive integer n. If R is a Euclidean ring, then it iswell-known that GL (R) GE (R) n n for every positive integer n. In 1966 (Publ. Math. IHES 30 (1966),5-53), P. M. Cohn introduced the concept of a generalized Euclidean ring, i.e., a ring R with unity is called ageneralized Euclidean ring, or GE-ring for short, if and only if GL (R) GE (R) n n for every positive integern. Recently in 2015, we prove that a ring R is a GE-ring if it is a quasi-Euclidean ring which is an anothergeneralization of the concept of a Euclidean ring. (Recall that a ring R is a quasi-Euclidean ring,introduced by B. Bougaut (1977) and A. Leutbecher (1978), if and only if it is a commutative ring with unityand every pair (b, a) of elements in R has a terminating division chain of finite length starting from it, the pair(b, a) with this property is also called a Euclidean pair by A. Alahmadi, S. K. Jain, T. Y. Lam, and A. Leroy(J. Algebra (2014)). As an example, let E be the ring of all algebraic integers in the field of complex numbers.Then E is a quasi-Euclidean ring, but it is not Euclidean.)The notion of the stable rank of a ring R, denoted by sr(R), was introduced by H. Bass in 1964. Theresults on the stable rank of rings have close relation to the concepts of GE-rings and n GE -rings. Forexample, if sr(R) = 1, then R is a GE-ring. As examples, sr(R) = 1 for every local ring R and every Artinianring R.In this project we will study the relations between the stable rank of rings and the concepts of GEringsand n GE -rings. We also want to identify more examples of rings which are quasi-Euclidean rings butnot Euclidean rings, to identify more examples of rings which are GE-rings but not quasi-Euclidean rings, toidentify more examples of rings which are n GE -rings for some integer n but not GE-rings. (Recall that aring R with unity is called a n GE -ring if GL (R) GE (R) n n for positive integer n.)
|Effective start/end date||1/08/17 → 31/07/18|
UN Sustainable Development Goals
In 2015, UN member states agreed to 17 global Sustainable Development Goals (SDGs) to end poverty, protect the planet and ensure prosperity for all. This project contributes towards the following SDG(s):
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